The Science of Life Lecture Series

So, there has been a debate going around social media as of late.  It deals with the order of operations, and the fact that the vast majority of people seem to forget that each step of the order of operations ends with the phrase "from left to right".  That phrase is irrelevant most of the time, but when it's not irreverent, it's crucial. Damn you PEMDAS! So what is 9!/8!(3^2)?  Is it 81 or 1?  Or is it something else entirely?  Now I am going to tread lightly here so as to not enrage too many hearts. Before I answer this question, it would be a good idea to review the algebraic order of operations , or PEMDAS: Perform all operations inside parentheses using the remainder of the order of operations as your guide to the order.  Any parentheses within parentheses are to be performed from the inside out and from left to right. Perform any and all exponent from left to right. Perform all multiplication and division from left to right. Perform all ad

Typically, problems involving at least two unknowns are solved by translating the problem to two or more equations, such as the slope-intercept form and/or the standard form of the line. Developing Equations from Sentence Form When deciphering word problems, we typically try to take a word or phrase which explains what a variable is and represent that word or phrase as a single letter instead. For instance, the energy consumed by your household so far this billing cycle can be represented by the letter E, and the total cost so far in that pay cycle can be represented by the letter C. It is also useful to translate commonly used words into mathematical operations. For example, the product of two things m and c represents multiplication, while the square of c represents squaring the letter c. So the product of the two values of m and the square of c would be $m \times c^2$. I will leave a table of some common and uncommon phrases and their mathematical translations here . Outs

The process of problem solving - commonly known as solving word problems - tends to give students of Algebra the most trouble. Part of the issue is the lack of intuitive understanding; a bigger issue is the lack of good explanations from both text books and teachers. There tends to be less explicit explanations available for for how to set up word problems to equation form than for any other section of Algebra. Once the issue of setting up problems is resolved, solving them becomes much easier. This chapter deals with the mechanism of setting up problems in order to solve them. This is necessary for all physical, life, and social sciences as well as all forms of accounting and economics. Basically, anything which is fundamentally numerical in nature tequires this skill. I will try my best to explain how to set up a word prolem to the best of my ability, so sit back, relax, and get ready to understand. Section 1: Systems of Equations in Two Equations Section 2: Solvin

When interpreting word problems as math equations, it is a good idea to keep in mind that certain phrases will always represent a certain mathematical operation.  Here is a list of some of these words for the basic operations. Addition Subtraction Multiplication Division Equals Exponent Parenthases Add Subtract Multiply Divide Is To the power of Quantity Sum Difference Product Quotient Yields Raised to the Increased by Decreased by Times As much Set as Plus Minus Double ($\times 2$) Halve ($\frac{x}{2}$) Equal to More Less Triple ($\times 3$) A third ($\frac{x}{3}$) Determined to be Combined Change Twice ($\

Hello internet, and welcome to the Algebra Lecture Series from the Science of Life.  This time, we are focusing on an introduction to the algebra of functions, or taking two or more functions and applying the algebraic functions in order to combine these functions. If we have two functions $f(x)$ and $g(x)$ which have the same domain, then applying the same x-value to them and performing a mathematical operation on them yields a single value.  It is important that they either have the same domain or have overlapping domain, since this is the only way that there would be a non-zero value for each function.  For example, a florist cannot physically store a negative quantity of flowers nor an amount of flowers greater than the physical space available.  Therefor, the domain is from 0 to some finite number determined by the physical space.  The same applies for other plants.  So flowers and plants have the same domain. One mechanism for applying an operation to the functions is to

Hello internet, and welcome to the Algebra Lecture Series from The Science of Life.  This entry is focusing a little deeper into the concept of lines, especially the other equations of the line. Suppose that we have a line with slope $m$ and it passes through a specific point $(x_{1}, y_{1})$.  For any other general point (x, y) on the line (where $x_{1} \neq x$ and $y_{1} \neq y$) on the line, the slope of the line would be $m=\frac{y-y_{1}}{x-x_{1}}$.  Multiplying both sides of the equation by the denominator of the right hand side gives us $m \times (x-x_{1})=(y-y_1)$.  This final form is called the point-slope form of the linear equation, since we use a point and the slope. The two green lines are parallel. If we look at two parallel lines, they do not cross (they have no solution) by the definition of parallel lines.  This can only be done if the change in the y variable (the rise) for every unit change in the x variable are exactly the same for both lines.  Notice

Hello internet, and welcome to the Algebra Lecture Series from The Science of Life.  This entry is an introduction to Linear Functions in both graphs and models. There are two situations where the slope would not be stated in the slope-intercept form.  These are cases where the line crosses only one of the 2 (or more) axes. First, let's consider the horizontal line.  If we have a line which has a y-intercept but does not have an x-intercept (does not cross the x-axis), it is called a horizontal line.  Let's look at it from the point of view of the slope-intercept form we developed last time, $y=mx+b$.  We know that the slope is the change in the y-value per change in the x-value.  If we have two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$, the slope is mathematically defined as $m=\frac{y2-y1}{x2-x1}$.  The denominator, the $ dx=x2-x1$, is never going to be zero when there's a y-intercept, since we can go from the left of the y-intercept to the right of the y-inter

Hello internet, and welcome to the Algebra Lecture Series from The Science of Life.  This entry is an introduction to Linear Functions in both graphs and models. Linear Equations are the easiest functions to conceptualize. They are lines which stretch out to infinity in both directions. On an x-y plane, the equation of a line can follow many forms, the first one usually introduced being the slope-intercept form y=mx+b, where m is the slope (change in the y-coordinate per 1 unit change in the x-coordinate) and b is the y-intercept (point where it crosses the y-axis). There are other forms of the equation of the line, but I'll get to those in future entries. If we have multiple points which form an obvious line, we can use two of those points to find the slope of the line. As mentioned earlier, the slope is the motion along the y-axis as the x-coordinate changes. With that in mind, we can use two points (x1, y1) and (x2, y2) to determine the slope. The slope is given by

Hello internet, and welcome to the Algebra Lecture Series from The Science of Life.  In this entry, we move to the definition of an algebraic function. I feel it is a good idea to start with the definitions of domain and range.  The domain is defined as the set of all possible values for which the independent variable (the input, the x-value) can take up.  This is all possible values for the horizontal (x) axis.  The range is the set of all possible values for which the dependent variable (the output, the y-value) can take up.  This is all possible values for the vertical (y) axis. The blue horizontal axis (typically the x-axis) is the domain and the independent variable. The green vertical axis (typically the y-axis) is the range and the dependent variable. Now we move to the concept of a function.  A good definition of the function is any equation where each input value has exactly one output value.  The technical version of this is "any equation where, for each value

If you're familiar with a number line, then you know that each point on that number line corresponds to a number.  If we take two number lines which are perpindicular to one another (they cross each other to make a right angle) and have them cross at the others zero-point, then we have a method of analyzing two-dimensional data-sets and equations, data and equations with two variables which relate to one another.  This two-number-line situation is typically called either the x-y coordinate system (because the horizontal number line is labeled as the x-number line and the horizontal is labeled as the y-number-line) or the Cartessian Coordinate System (named after French mathematician René Descartes, supposedly the first person to come up with the system).  The two number lines, since they're in the context of a coordinate system, are each now called an axis (plural: axes). The x-axis and y-axis of a Cartesian coordinate system. Any point on this coordinate system is give